Parkin scored the Grecians' first away goal of the season with 10 minutes remaining to cancel out Josh Scott's opener for the hosts.
The Daggers bossed proceedings in the first half and deservedly led at the break.
A howler by Exeter goalkeeper Artur Krysiak led to Scott scoring in the 20th minute. Coming to meet Billy Bingham's hopeful ball forward, Krysiak mistimed his clearance and allowed Scott to pounce on the loose ball before hooking his shot into an unguarded net from 25 yards.
Krysiak then saved Abu Agogo's effort before Matt Grimes blazed over for Exeter a minute before the break.
Zavon Hines was a constant threat for Dagenham and his shot was well saved by Krysiak shortly after the restart.
However, Exeter had the better of the closing stages and levelled when 32-year-old striker Parkin headed in his second goal of the season from Matt Oakley's cross.